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September  2011, 10(5): 1393-1400. doi: 10.3934/cpaa.2011.10.1393

Periodic solutions for $p$-Laplacian systems of Liénard-type

Wenbin Liu 1, and  Zhaosheng Feng 2,

1. 

Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China
2. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  February 2009 Revised  July 2010 Published  April 2011

In this paper, we study the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems by means of the topological degree theory. Sufficient conditions of the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems of Liénard-type are presented.
Keywords: topological degree, Liénard equation., existence, $p$--Laplacian system, periodic solution.
Mathematics Subject Classification: Primary: 34A12; Secondary: 34C2.
Citation: Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
References:
[1]

L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, "Geography and Geographysics IX," 7-10 (In Russian), 1983. Google Scholar

[2]

L. E. Bobisud, Steady state turbulent flow with reaction, Rochy Mountain J. Math., 21 (1991), 993-1007.  Google Scholar

[3]

J. D. Murray, "Mathematical Biology," Springer-Verlag, New York, 1993.  Google Scholar

[4]

M. A. Herrero and J. L. Vazquz, On the propagation properties of a nonlinear degenerate parabolic equation, Commun. Partial Diff. Equs., 7 (1982), 1381-1402.  Google Scholar

[5]

L. Boccardo, P. Drábek, D. Giachetti and M. Kučera, Generalization of Fredholm alternative for nonlinear differential operators, Nonlin. Anal., 10 (1986), 1083-1103.  Google Scholar

[6]

C. Coster, On pairs of positive solutions for the one-dimensional p-Laplacian, Nonlin. Anal., 23 (1994), 669-681.  Google Scholar

[7]

M. Del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0, u(0)=u(T)=0, p>1$, J. Diff. Equs., 80 (1989), 1-13.  Google Scholar

[8]

R. Manásevich and F. Zanolin, Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian, Nonlin. Anal., 21 (1993), 269-291.  Google Scholar

[9]

M. Zhang, Nonuniform nonresonance at the first eigenvalue of the p-Laplacian, Nonlin. Anal., 29 (1997), 41-51.  Google Scholar

[10]

M. Del Pino, R. Manásevich and A. Murua, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e., Nonlin. Anal., 18 (1992), 79-92.  Google Scholar

[11]

M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc., 64 (2001), 125-143.  Google Scholar

[12]

P. Amster and P. De Nápoli, Landesman-Lazer type conditions for a system of p-Laplacian like operators, J. Math. Anal. Appl., 326 (2007), 1236-1243.  Google Scholar

[13]

M. Carcía-huidobro, C. P. Gupta and R. Manásevich, Solvability for a nonlinear three-point boundary value problem with p-Laplac-like operator at resonance, Abstr. Appl. Anal., 16 (2001), 191-214.  Google Scholar

[14]

R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p-$Laplacian-like operators, J. Diff. Equs., 145 (1998), 367-393.  Google Scholar

[15]

H. Liu and Z. Feng, Begehr-Hile operator and its applications to some differential equations, Commun. Pure Appl. Anal., 9 (2010), 387-395.  Google Scholar

[16]

S. B. Li, Y. H. Su and Z. Feng, Positive solutions to $p$-Laplacian multi-point BVPs on time scales, Dyn. Partial Differ. Equ., 7 (2010), 45-64.  Google Scholar

show all references

References:
[1]

L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, "Geography and Geographysics IX," 7-10 (In Russian), 1983. Google Scholar

[2]

L. E. Bobisud, Steady state turbulent flow with reaction, Rochy Mountain J. Math., 21 (1991), 993-1007.  Google Scholar

[3]

J. D. Murray, "Mathematical Biology," Springer-Verlag, New York, 1993.  Google Scholar

[4]

M. A. Herrero and J. L. Vazquz, On the propagation properties of a nonlinear degenerate parabolic equation, Commun. Partial Diff. Equs., 7 (1982), 1381-1402.  Google Scholar

[5]

L. Boccardo, P. Drábek, D. Giachetti and M. Kučera, Generalization of Fredholm alternative for nonlinear differential operators, Nonlin. Anal., 10 (1986), 1083-1103.  Google Scholar

[6]

C. Coster, On pairs of positive solutions for the one-dimensional p-Laplacian, Nonlin. Anal., 23 (1994), 669-681.  Google Scholar

[7]

M. Del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0, u(0)=u(T)=0, p>1$, J. Diff. Equs., 80 (1989), 1-13.  Google Scholar

[8]

R. Manásevich and F. Zanolin, Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian, Nonlin. Anal., 21 (1993), 269-291.  Google Scholar

[9]

M. Zhang, Nonuniform nonresonance at the first eigenvalue of the p-Laplacian, Nonlin. Anal., 29 (1997), 41-51.  Google Scholar

[10]

M. Del Pino, R. Manásevich and A. Murua, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e., Nonlin. Anal., 18 (1992), 79-92.  Google Scholar

[11]

M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc., 64 (2001), 125-143.  Google Scholar

[12]

P. Amster and P. De Nápoli, Landesman-Lazer type conditions for a system of p-Laplacian like operators, J. Math. Anal. Appl., 326 (2007), 1236-1243.  Google Scholar

[13]

M. Carcía-huidobro, C. P. Gupta and R. Manásevich, Solvability for a nonlinear three-point boundary value problem with p-Laplac-like operator at resonance, Abstr. Appl. Anal., 16 (2001), 191-214.  Google Scholar

[14]

R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p-$Laplacian-like operators, J. Diff. Equs., 145 (1998), 367-393.  Google Scholar

[15]

H. Liu and Z. Feng, Begehr-Hile operator and its applications to some differential equations, Commun. Pure Appl. Anal., 9 (2010), 387-395.  Google Scholar

[16]

S. B. Li, Y. H. Su and Z. Feng, Positive solutions to $p$-Laplacian multi-point BVPs on time scales, Dyn. Partial Differ. Equ., 7 (2010), 45-64.  Google Scholar

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